Is there a name for a matrix which is a multiple of an orthogonal matrix? I.e. a square matrix $A$ which satisfies the condition

$$A^TA = AA^T = \lambda I$$

where $\lambda$ is some scalar (which should perhaps be required to be non-zero) and $I$ is the identity matrix. Or in other words, a matrix whose rows and columns are orthogonal vectors of equal length $\sqrt\lambda$, but not neccessarily unit length, so not orthonormal.

I've thought about these objects twice in different contexts recently, and it feels like a concept that should have a name. So far I haven't been able to find such a name, though. Do you know an established name for these matrices?


In a (now deleted) comment, user104254 suggested the name

Conformal matrices

Actually he suggested the keyword “conformal”, the combination was assumed by me. That name is suppoerted by some papers on the web.

Conformal Matrices by Jeffrey Rauch in Corollary 4.2 uses the condition

$$M\,M^t=c^2\,I\qquad\text{with }M\text{ invertible and }c>0$$

and states in Definition 4.3 that matrices $M$ satisfying this equation are called conformal.

In Geometric Rigidity of Conformal Matrices by Daniel Faraco and Xiao Zhong, they define conformal matrices as

$$CO_+(n)=\{A\in M^{n\times n}:A=\rho R\text{, where }\rho\in\mathbb R_+\text{ and }R\in SO(n)\}$$

As far as I can tell, the two definitions are equivalent to one another and to my own statemement, at least over the reals. Both do explicitely exclude the case of $\lambda=0$.

  • $\begingroup$ Thank you for referring to me in your own answer. Upvoted. $\endgroup$
    – user104254
    Mar 11 '14 at 15:02
  • $\begingroup$ This could be seen as a special case of the more general conformal transformation, where $M M^t = c^2 g$, for some symmetric $g$. This is usually done when $g$ is a metric, although it's quite common in numerical treatments of general relativity, in which $g$ wouldn't be positive-definite. $\endgroup$
    – Muphrid
    Mar 11 '14 at 15:28

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